In this paper, we present a new concise approach for normalizing
m-photon-added (-subtracted) STS and for deriving the PNDs, which improve
the method used in Refs. [1]}, [2]}. That is to say, using the thermal
field dynamics theory, we convert the thermal state to a pure state in
doubled Fock space in which the calculations of ensemble averages under a
mixed state \(\rho \) , \(\left\langle A\right\rangle =\mathtt {tr}\left( A\rho \right) /\mathtt {tr}\left( \rho \right) \ \) is replaced by an equivalent
expectation values with a pure state \(\left|0(\beta )\right\rangle \) ,
i.e., \(\left\langle A\right\rangle =\left\langle 0(\beta )\right|A\left|0(\beta )\right\rangle \) . It is shown that the average value of \(e^{fa^{\dag }a}\)  under STS is just the generating function of Legendre
polynomials, a remarkable result. Based on this point, the normalization and
PNDs of m-photon added (or subtracted) STS are easily obtained as the
Legendre polynomials. The generating function of the Legendre polynomials
and the average value of \(e^{fa^{\dag }a}\)  under STS are used in the whole
calculation.
